Lecture 2: Z-TransformXILIANG LUO

2014/9

Fourier Transform

Convergence A sufficient condition: absolutely summable

it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function

Square Summable A sequence is square summable if:

For square summable sequence, we have mean-square convergence:

∑𝑛=−∞

∞

|𝑥 [𝑛 ]|2<∞

Z-Transform

a function of the complex variable: z

If we replace the complex variable z by , we have the Fourier Transform!

Z-Transform & Fourier Transform

Complex z-plane

Region of Convergence The set of z for which the z-transform converges is called ROC of the z-transform.

Absolutely summable criterion:

ROC ROC consists of a ring in the z-plane

Closed-Form in ROC When X(z) is a rational function inside ROC, i.e.

P(z), Q(z) are polynomials in z Zeros: values of z such that X(z) = 0 Poles: values of z such that X(z) = infinity

Z-Transform Example: Right-Sided

Z-Transform Example:Left-Sided

Diff. Sum, Same Z-Transform? One is right-sided exponential sequence

One is left-sided exponential sequence

But they share the same algebraic expressions for their Z-Transforms

This emphasizes the importance of the region of convergence!!

ROC Properties

ROC Properties

ROC Properties

Inverse z-Transform From the z-Transform, we can recover the original sequence using the following complex contour integral:

𝑥 [𝑛 ]= 12𝜋 𝑗∮𝐶

❑

𝑋 (𝑧 )𝑧𝑛−1𝑑𝑧

C is a closed contour within the ROC of the z-transform

Inverse z-Transform Methods Inspection

familiar with the common transform pairs

Partial Fraction Expansion

Power Series Expansion

z-Transform Properties 1. Linearity

2. Time Shifting

3. Multiplication by an Exponential Sequence

z-Transform Properties 4. Differentiation of X(z)

5. Conjugation of a Complex Sequence

7. Time Reversal

z-Transform Properties 7. Convolution of Sequences

z-Transform and LTI Systems LTI system is characterized by its impulse response h[n]

h[n]x[n] y[n]

𝑦 [𝑛 ]=𝑥 [𝑛 ]⋆h[𝑛]

𝑌 (𝑧 )=𝑋 (𝑧 )×𝐻 (𝑧 )

H(z) is called the system function of this LTI system!

Cauchy-Riemann Equations If function f(z) is differentiable at z0=x0+y0, then its component functions must satisfy the following conditions:

𝑓 (𝑧 )=𝑢 (𝑥 , 𝑦 )+𝑖𝑣 (𝑥 , 𝑦)

𝜕𝑢𝜕𝑥

=𝜕 𝑣𝜕 𝑦

𝜕𝑢𝜕 𝑦

=−𝜕 𝑣𝜕 𝑥

Analytic Functions A function f(z) is analytic at a point z0 if it has a derivative at each point in some neighborhood of z0.

So, If f(z) is analytic at a point z0, it must be analytic at each point in some neighborhood of z0.

Taylor Series Theorem: Suppose that a function f is analytic throughout a disk: |z-z0|<R0, centered at z0 and with radius R0, then f(z) has the power series representation:

𝑓 (𝑧 )=∑𝑛=0

+∞

𝑎𝑛 (𝑧 −𝑧0 )𝑛 |𝑧− 𝑧0|<𝑅0

𝑎𝑛=𝑓 (𝑛 )(𝑧0)𝑛 !

Laurent Series If a function is not analytic at a point z0, one cannot apply Taylor’s theorem at that point!

Laurent’s Theorem: Suppose a function f is analytic throughout an annular domain centered at z0:

Let C denote any positively oriented simple closed contour around z0 and lying in the domain, then, at each point in the domain, f(z) has the series representation:

𝑅1<|𝑧−𝑧 0|<𝑅2

𝑓 (𝑧 )= ∑𝑛=−∞

+∞

𝑐𝑛 (𝑧−𝑧 0 )𝑛

Laurent Series

𝑐𝑛=12𝜋 𝑖∮𝐶

❑ 𝑓 (𝑧 )𝑑𝑧

(𝑧− 𝑧0 )𝑛+1

𝑓 (𝑧 )= ∑𝑛=−∞

+∞

𝑐𝑛 (𝑧−𝑧 0 )𝑛

Homework Problems

3.59:

3.57:

3.52:

3.56:

Next Sampling of Continuous-Time Signals

Please read the textbook Chapter 4 in advance!